# On the solutions of linear Volterra equations of the second kind with sum kernels

We consider a linear Volterra integral equation of the second kind with a sum kernel K(t',t)=∑_i K_i(t',t) and give the solution of the equation in terms of solutions of the separate equations with kernels K_i, provided these exist. As a corollary, we obtain a novel series representation for the solution with much improved convergence properties. The error resulting from truncation of this series can be made as small as desired at any truncation order in a trade-off with increased computational complexity upon tuning a certain parameter. As a byproduct, we obtain a novel product integral representation of the solution with respect to a convolution-like product. We illustrate our results with examples, including the first known Volterra equation solved by Heun's confluent functions. This solves a long-standing problem pertaining to the representation of such functions. The approach presented here has widespread applicability in physics via Volterra equations with degenerate kernels.

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